Complex Numbers And Quadratic Equations question 434
Question: If $ \sum\limits_{k=0}^{100}{i^{k}}=x+iy $ , then the values of $ x $ and $ y $ are
Options:
A) $ x=-1,y=0 $
B) $ x=1,y=1 $
C) $ x=1,y=0 $
D) $ x=0,y=1 $
Show Answer
Answer:
Correct Answer: C
Solution:
$ \sum\limits_{k=0}^{100}{i^{k}=x+iy,} $
Þ $ 1+i+i^{2} $ $ +……+i^{100}=x+iy $ Given series is G.P.
Þ $ \frac{1.(1-i^{101})}{1-i}=x+iy $
Þ $ \frac{1-i}{1-i}=x+iy $
Þ $ 1+0i=x+iy $ Equating real and imaginary parts, we get the required result.
BETA
